On Gauss-Bonnet Curvatures

Mohammed Larbi LabbiMathematics Department, College of Science, University of Bahrain, 32038 Bahrain

Received August 27, 2007, in final form November 15, 2007; Published online December 11, 2007

Abstract
The (2k)-th Gauss-Bonnet curvature is a generalization to higher dimensions
of the (2k)-dimensional Gauss-Bonnet integrand, it coincides with the usual scalar curvature for k =1.
The Gauss-Bonnet curvatures are used in theoretical
physics to describe gravity in higher dimensional
space times where they are known as the Lagrangian of Lovelock gravity, Gauss-Bonnet Gravity and Lanczos gravity.
In this paper we present various aspects of these curvature invariants and review their
variational properties. In particular, we discuss natural generalizations of the Yamabe
problem, Einstein metrics and minimal submanifolds.